If the wires running from the main breaker panel are big enough the circuit could be switched over to 220v by just changing the breaker. If there are three wires two can be hot and one neutral with a ground rod outside for grounding. Then a sub-panel that would give 220v for a lathe and 110v for lighting and outlets could be used.
What you are getting at is to use the ground wire (bare) in 3 conductor Romex as Neutral, then Black and White as L1 and L2 (your two "hot" lines). The convention with 110v is that Black is Hot, White is Neutral, and Bare is Ground. For switching to 220v you could set Black to L1 and White to L2 (your two hots) then keep Bare as ground. Not having Neutral means you would have no 110v. Loading up the Bare wire as Neutral doesn't seem like a "hot" idea even if you use a "ground rod".
Incidentally, Neutral and Ground are bonded in most main breaker panels, however Neutral is still the line that is to complete the AC circuit for 110v power, not the bare ground wire. Sub-panels are more commonly set up with separate ground and neutral (that gets in to a complicated mess with out-buildings and tying in generators - no generator company will take the liability of separating neutral and ground (they bond neutral of the alternator to the frame) so when you feed a floating panel the ground and neutral are automatically connected at the generator. Of course, people modify stuff to get around that. But to each their own and to each their own risk.
Getting 220 isn't going to be easy. I rent, the garage is detached and the power to the garage is run under the driveway. The house has 220, but this wouldn't be a cheap project.
I rented a place that had an out-building/garage that didn't have power. It did at one point, but not when I got there. There was a 20 amp 110v circuit with its own outlet at the main breaker panel. So I ran a 12g romex "extension cord" from that outlet to the building. All said it was 225ft or so. There was a 200ft run outside plus the bit inside to get to the outlet. 12g romex was not adequate for the set up. For lights it was OK, light power tools (hand tools) it was OK past start-up, and when my 20gal air compressor tried to start up everything would shut off with the start up load but it would go - slowly. I don't recall any welding projects while I was there, but that would have been the worst idea - the voltage drop with current would cause unsteady arc power, and a light arc at that.
I would venture to guess that your out-building isn't 200ft from the house, so that is a good thing.
If I were in your position - I would do the extension cord idea. The run I had was in flex conduit outside. Pulling the wire through the conduit was a real bugger. The conduit was 2x 100ft runs. I ran small PVC pipe with a string threaded all the way through. I started at one end and pushed the PVC through, added another piece, and kept going until the PVC came out the other end. Then I used the string to pull the romex.
Use NMEA L14-30 plugs and run 4 conductors (both hots, neutral, and ground). The normal NMEA 14-30 dryer style plug doesn't twist lock like the L14-30 - it is OK to hang on the wall, but if you bump a long cord and pull on it the locking plugs are a lot better for that reason (and they are the normal 120/240v outlet on generators up to about 8000w).
I suppose if you already have 110v for lights etc out there you wouldn't need neutral in an "extension cord" - then just run both hots and ground for 220v. However, if you are doing the effort you might as well add a wire, make 4 conductors, and use a 4 pin plug for both 120/240v. I wish I did that when I made my extension cord, just to have it. At the moment its running power to an out-building at a property in the family - the same 200ft or so length - for lights and tools.
The idea of converting the circuit that is there to 220v just doesn't sit well with me no matter how I look at it. It might "work", and I can see the logic in it. Electricians (licensed) may say it will (I have no idea - I'm not a licensed electrician, just speaking in open terms here). It just doesn't sound like a good idea to me.
